Question

Solve the following question in detail:
On the set \[S\] of all real numbers, define a relation \[R = \left\{ {\left( {a,b} \right):a \leqslant b} \right\}\]. Show that \[R\] is reflexive.

Answer 1

Hint: Consider the given condition. To prove that it is a reflexive property, first identify and explain what reflexive property is. Take two values assigned to the variables and see if there is any matching to the definition and explanation of the given variables. Then substitute the values and check if they are either reflexive or if they suit any other property.

Complete step-by-step solution:
Given,
\[R = \left\{ {\left( {a,b} \right):a \leqslant b} \right\}\]
We have to prove that it is a reflexive property.
We are given two variables.
Let us take values for each variable and prove that they are reflexive.
So,
When \[a = 1\], we can have two values for \[b\] because of the stated condition.
\[ \Rightarrow b = 1,2\]
The ordered pairs are as follows which are possible.
\[\left( {1,2} \right),\left( {1,1} \right),\left( {2,2} \right)\]
We have the two elements of the ordered pair same.
It is reflexive. So there is no need to check if it is transitive or symmetrical because the question asked is to prove if they are reflexive.

\[\therefore R\] is reflexive.

Note: The reflexive property in mathematics states that any real number, let us consider \[a\], is equal to itself. Also written as, \[a = a\]. We also have two other properties to check the sets of any given condition. Symmetric property and transitive property.
Symmetric property states that, for any real numbers, let us consider two, \[a\] and \[b\], we have \[a = b\]
Then moving onto the transitive property, it states that, for any real numbers, let us consider \[a,b\] and \[c\], if \[a = b,b = c\] and \[c = a\], it is said to be transitive.